Commutative property

A binary operation is commutative if changing the order of its two operands never changes the result, a property familiar from ordinary arithmetic facts such as three plus four equalling four plus three, or two times five equalling five times two. Not every operation has this property: subtraction and division do not, since for example three minus five does not equal five minus three, and operations lacking the property are called noncommutative. Although the commutativity of simple arithmetic operations was implicitly assumed for centuries, the property itself was not given a name until the nineteenth century, when mathematicians began studying new algebraic structures in which it could no longer be taken for granted.